document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199899>Question 1199899</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a house of twenty people ,8 liked Ampesi,12 liked banku and 12 liked Fufu.5 liked both Ampesi <BR>\n" );
document.write( "and Fufu,6 liked Banku and Fufu,2 only Ampesi and 3 liked all three types of food.<BR>\n" );
document.write( "a). Illustrate the information on a Venn diagram<BR>\n" );
document.write( "b). Find the number of in the house who liked ;<BR>\n" );
document.write( "(i). only Fufu<BR>\n" );
document.write( "(ii). Exactly two types<BR>\n" );
document.write( "(iii).Only one type of food. </SPAN>\n" );
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document.write( "I'll get you started with part (a).\r<BR>\n" );
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document.write( "Draw a rectangle to represent the universal set.<BR>\n" );
document.write( "Inside the rectangle will be 3 partially overlapped circles as shown below.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C300%2C-10%2C10%2C-10%2C10%2C%0D%0Acircle%28-2%2C2%2C4%29%2C%0D%0Acircle%282%2C2%2C4%29%2C%0D%0Acircle%280%2C-2%2C4%29%2C%0D%0Alocate%28-4%2C4.8%2C%22A%22%29%2C%0D%0Alocate%280%2C4.8%2C%22B%22%29%2C%0D%0Alocate%284%2C4.8%2C%22C%22%29%2C%0D%0Alocate%28-2.5%2C-1%2C%22D%22%29%2C%0D%0Alocate%280%2C1%2C%22E%22%29%2C%0D%0Alocate%282.5%2C-1%2C%22F%22%29%2C%0D%0Alocate%280%2C-5%2C%22G%22%29%2C%0D%0Alocate%28-5%2C-6%2C%22H%22%29%2C%0D%0Aline%28-8%2C8%2C8%2C8%29%2C%0D%0Aline%288%2C8%2C8%2C-8%29%2C%0D%0Aline%288%2C-8%2C-8%2C-8%29%2C%0D%0Aline%28-8%2C-8%2C-8%2C8%29%2C%0D%0Alocate%28-7%2C7%2C%22Ampesi%22%29%2C%0D%0Alocate%285%2C7%2C%22Banku%22%29%2C%0D%0Alocate%283%2C-5.5%2C%22Fufu%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The 8 regions A,B,C,D,E,F,G,H represent the different scenarios.<BR>\n" );
document.write( "For example, if someone is in region A, that person likes Ampesi only (they dont like Banku or Fufu).<BR>\n" );
document.write( "This region is in the \"Ampesi\" circle but outside the other circles.\r<BR>\n" );
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document.write( "Another example: If you are in region E, then you like all three food types. This region is found in all three circles.\r<BR>\n" );
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document.write( "Here are the given facts:<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Number</TD><TD >Statement</TD></tr><tr><TD ><font color=blue>1</font></TD><TD >House of twenty people</TD></tr><tr><TD ><font color=blue>2</font></TD><TD >8 liked Ampesi</TD></tr><tr><TD ><font color=blue>3</font></TD><TD >12 liked Banku</TD></tr><tr><TD ><font color=blue>4</font></TD><TD >12 liked Fufu</TD></tr><tr><TD ><font color=blue>5</font></TD><TD >5 liked both Ampesi and Fufu</TD></tr><tr><TD ><font color=blue>6</font></TD><TD >6 liked Banku and Fufu</TD></tr><tr><TD ><font color=blue>7</font></TD><TD >2 [liked] only Ampesi</TD></tr><tr><TD ><font color=blue>8</font></TD><TD >3 liked all three types of food.</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "<font color=blue>Fact 8</font> leads directly to E = 3<BR>\n" );
document.write( "In other words, we replace the \"E\" in the diagram above with the value 3, since 3 people like all of the food types mentioned.\r<BR>\n" );
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document.write( "<font color=blue>Fact 5</font> says \"5 liked both Ampesi and Fufu\"<BR>\n" );
document.write( "Use this in conjunction with <font color=blue>fact 8</font>.<BR>\n" );
document.write( "If 5 liked both Ampesi and Fufu, and 3 liked all the food types mentioned, then 5-3 = 2 people liked Ampesi and Fufu only (and not Banku).\r<BR>\n" );
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document.write( "In short, we write \"2\" in region D.\r<BR>\n" );
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document.write( "Now use <font color=blue>fact 6</font> and <font color=blue>fact 8</font> together to conclude that 6-3 = 3 people liked Banku and Fufu only (but not Ampesi). The value 3 goes in region F.\r<BR>\n" );
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document.write( "Let's update the diagram with what we know so far.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C300%2C-10%2C10%2C-10%2C10%2C%0D%0Acircle%28-2%2C2%2C4%29%2C%0D%0Acircle%282%2C2%2C4%29%2C%0D%0Acircle%280%2C-2%2C4%29%2C%0D%0Alocate%28-4%2C4.8%2C%22A%22%29%2C%0D%0Alocate%280%2C4.8%2C%22B%22%29%2C%0D%0Alocate%284%2C4.8%2C%22C%22%29%2C%0D%0Alocate%28-2.5%2C-1%2C%222%22%29%2C%0D%0Alocate%280%2C1%2C%223%22%29%2C%0D%0Alocate%282.5%2C-1%2C%223%22%29%2C%0D%0Alocate%280%2C-5%2C%22G%22%29%2C%0D%0Alocate%28-5%2C-6%2C%22H%22%29%2C%0D%0Aline%28-8%2C8%2C8%2C8%29%2C%0D%0Aline%288%2C8%2C8%2C-8%29%2C%0D%0Aline%288%2C-8%2C-8%2C-8%29%2C%0D%0Aline%28-8%2C-8%2C-8%2C8%29%2C%0D%0Alocate%28-7%2C7%2C%22Ampesi%22%29%2C%0D%0Alocate%285%2C7%2C%22Banku%22%29%2C%0D%0Alocate%283%2C-5.5%2C%22Fufu%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now turn to <font color=blue>fact 4</font><BR>\n" );
document.write( "12 people like Fufu<BR>\n" );
document.write( "The values in the \"Fufu\" circle must add to 12<BR>\n" );
document.write( "G+2+3+3 = 12<BR>\n" );
document.write( "G+8 = 12<BR>\n" );
document.write( "G = 12-8<BR>\n" );
document.write( "G = 4\r<BR>\n" );
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document.write( "Next we look at <font color=blue>fact 7</font>.<BR>\n" );
document.write( "2 people like Ampesi only.<BR>\n" );
document.write( "Update the value \"A\" to have it replaced with 2.\r<BR>\n" );
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document.write( "Update so far:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C300%2C-10%2C10%2C-10%2C10%2C%0D%0Acircle%28-2%2C2%2C4%29%2C%0D%0Acircle%282%2C2%2C4%29%2C%0D%0Acircle%280%2C-2%2C4%29%2C%0D%0Alocate%28-4%2C4.8%2C%222%22%29%2C%0D%0Alocate%280%2C4.8%2C%22B%22%29%2C%0D%0Alocate%284%2C4.8%2C%22C%22%29%2C%0D%0Alocate%28-2.5%2C-1%2C%222%22%29%2C%0D%0Alocate%280%2C1%2C%223%22%29%2C%0D%0Alocate%282.5%2C-1%2C%223%22%29%2C%0D%0Alocate%280%2C-5%2C%224%22%29%2C%0D%0Alocate%28-5%2C-6%2C%22H%22%29%2C%0D%0Aline%28-8%2C8%2C8%2C8%29%2C%0D%0Aline%288%2C8%2C8%2C-8%29%2C%0D%0Aline%288%2C-8%2C-8%2C-8%29%2C%0D%0Aline%28-8%2C-8%2C-8%2C8%29%2C%0D%0Alocate%28-7%2C7%2C%22Ampesi%22%29%2C%0D%0Alocate%285%2C7%2C%22Banku%22%29%2C%0D%0Alocate%283%2C-5.5%2C%22Fufu%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<font color=blue>Fact 2</font> says \"8 liked Ampesi\"<BR>\n" );
document.write( "Meaning the values in the \"Ampesi\" circle must add to 8<BR>\n" );
document.write( "A+B+D+E = 8<BR>\n" );
document.write( "2+B+2+3 = 8<BR>\n" );
document.write( "B+7 = 8<BR>\n" );
document.write( "B = 8-7<BR>\n" );
document.write( "B = 1<BR>\n" );
document.write( "One person likes Ampesi and Banku only, but does not like Fufu.\r<BR>\n" );
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document.write( "\n" );
document.write( "Another update:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C300%2C-10%2C10%2C-10%2C10%2C%0D%0Acircle%28-2%2C2%2C4%29%2C%0D%0Acircle%282%2C2%2C4%29%2C%0D%0Acircle%280%2C-2%2C4%29%2C%0D%0Alocate%28-4%2C4.8%2C%222%22%29%2C%0D%0Alocate%280%2C4.8%2C%221%22%29%2C%0D%0Alocate%284%2C4.8%2C%22C%22%29%2C%0D%0Alocate%28-2.5%2C-1%2C%222%22%29%2C%0D%0Alocate%280%2C1%2C%223%22%29%2C%0D%0Alocate%282.5%2C-1%2C%223%22%29%2C%0D%0Alocate%280%2C-5%2C%224%22%29%2C%0D%0Alocate%28-5%2C-6%2C%22H%22%29%2C%0D%0Aline%28-8%2C8%2C8%2C8%29%2C%0D%0Aline%288%2C8%2C8%2C-8%29%2C%0D%0Aline%288%2C-8%2C-8%2C-8%29%2C%0D%0Aline%28-8%2C-8%2C-8%2C8%29%2C%0D%0Alocate%28-7%2C7%2C%22Ampesi%22%29%2C%0D%0Alocate%285%2C7%2C%22Banku%22%29%2C%0D%0Alocate%283%2C-5.5%2C%22Fufu%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Use <font color=blue>fact 3</font> to find that...<BR>\n" );
document.write( "B+C+E+F = set of twelve people who like Banku<BR>\n" );
document.write( "1+C+3+3 = 12<BR>\n" );
document.write( "C+7 = 12<BR>\n" );
document.write( "C = 12-7<BR>\n" );
document.write( "C = 5<BR>\n" );
document.write( "Five people like Banku only. They do not like the other two food types.\r<BR>\n" );
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document.write( "Another update:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C300%2C-10%2C10%2C-10%2C10%2C%0D%0Acircle%28-2%2C2%2C4%29%2C%0D%0Acircle%282%2C2%2C4%29%2C%0D%0Acircle%280%2C-2%2C4%29%2C%0D%0Alocate%28-4%2C4.8%2C%222%22%29%2C%0D%0Alocate%280%2C4.8%2C%221%22%29%2C%0D%0Alocate%284%2C4.8%2C%225%22%29%2C%0D%0Alocate%28-2.5%2C-1%2C%222%22%29%2C%0D%0Alocate%280%2C1%2C%223%22%29%2C%0D%0Alocate%282.5%2C-1%2C%223%22%29%2C%0D%0Alocate%280%2C-5%2C%224%22%29%2C%0D%0Alocate%28-5%2C-6%2C%22H%22%29%2C%0D%0Aline%28-8%2C8%2C8%2C8%29%2C%0D%0Aline%288%2C8%2C8%2C-8%29%2C%0D%0Aline%288%2C-8%2C-8%2C-8%29%2C%0D%0Aline%28-8%2C-8%2C-8%2C8%29%2C%0D%0Alocate%28-7%2C7%2C%22Ampesi%22%29%2C%0D%0Alocate%285%2C7%2C%22Banku%22%29%2C%0D%0Alocate%283%2C-5.5%2C%22Fufu%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We're one step away from finishing.\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "<font color=blue>Fact 1</font> says there are 20 people total.<BR>\n" );
document.write( "The values in the circles add to:<BR>\n" );
document.write( "A+B+C+D+E+F+G = 2+1+5+2+3+3+4 = 20<BR>\n" );
document.write( "showing that there are 20-20 = 0 people outside of the circles. Meaning no one person disliked all of the food types mentioned. All 20 people are somewhere in at least one circle.\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "Here is the final Venn Diagram<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C300%2C-10%2C10%2C-10%2C10%2C%0D%0Acircle%28-2%2C2%2C4%29%2C%0D%0Acircle%282%2C2%2C4%29%2C%0D%0Acircle%280%2C-2%2C4%29%2C%0D%0Alocate%28-4%2C4.8%2C%222%22%29%2C%0D%0Alocate%280%2C4.8%2C%221%22%29%2C%0D%0Alocate%284%2C4.8%2C%225%22%29%2C%0D%0Alocate%28-2.5%2C-1%2C%222%22%29%2C%0D%0Alocate%280%2C1%2C%223%22%29%2C%0D%0Alocate%282.5%2C-1%2C%223%22%29%2C%0D%0Alocate%280%2C-5%2C%224%22%29%2C%0D%0Alocate%28-5%2C-6%2C%220%22%29%2C%0D%0Aline%28-8%2C8%2C8%2C8%29%2C%0D%0Aline%288%2C8%2C8%2C-8%29%2C%0D%0Aline%288%2C-8%2C-8%2C-8%29%2C%0D%0Aline%28-8%2C-8%2C-8%2C8%29%2C%0D%0Alocate%28-7%2C7%2C%22Ampesi%22%29%2C%0D%0Alocate%285%2C7%2C%22Banku%22%29%2C%0D%0Alocate%283%2C-5.5%2C%22Fufu%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This concludes part (a).\r<BR>\n" );
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document.write( "This step-by-step process might seem a bit lengthy. However, once you get the hang of the basic arithmetic that's going on, it should come second nature to you. \r<BR>\n" );
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document.write( "Use that Venn Diagram to answer the other questions. I'll let the student take over from here.\r<BR>\n" );
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document.write( "Here is another problem involving a Venn Diagram of 3 circles.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.algebra.com/algebra/homework/Subset/Subset.faq.question.1196992.html\">https://www.algebra.com/algebra/homework/Subset/Subset.faq.question.1196992.html</a><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );